{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# American options\n",
    "\n",
    "\n",
    "## Contents\n",
    "   - [Optimal early exercise boundary](#sec1)\n",
    "   - [Binomial tree](#sec2)\n",
    "   - [Longstaff - Schwartz](#sec3)\n",
    "   - [Perpetual put](#sec4)\n",
    "   \n",
    "   \n",
    "The American option problem is an optimal stopping problem formulated as follows:\n",
    "\n",
    "$$ V(t,s) =  \\sup_{\\tau \\in [t,T]} \\mathbb{E}^{\\mathbb{Q}}\\biggl[ e^{-r(\\tau-t)} \\Phi(S_{\\tau}) \\bigg| S_t=s \\biggr]$$\n",
    "\n",
    "where $\\Phi(\\cdot)$ is the payoff, also called *intrinsic value*.    \n",
    "This formula resembles the formula of a European option, except that in this case the option can be exercised at any time. But... at which time? The problem is to find the best exercise time $\\tau$ that maximizes the expectation.\n",
    "\n",
    "Notice that the time $\\tau$ is a random variable! It can be different for different paths of the stock process.   \n",
    "The solution of the problem provides a strategy, that at each time assesses if it is optimal to exercise the option or not, depending on the current value of the underlying stock.\n",
    "\n",
    "It can be proved that the function $V(t,s)$ is the solution of the following PDE:\n",
    "\n",
    "\\begin{equation}\n",
    "\\max \\biggl\\{  \\frac{\\partial  V(t,s)}{\\partial t}  \n",
    "          + r\\,s \\frac{\\partial V(t,s)}{\\partial s}\n",
    "          + \\frac{1}{2} \\sigma^2 s^2 \\frac{\\partial^2  V(t,s)}{\\partial s^2} - r  V(t,s), \\;  \n",
    "          \\Phi(s) - V(t,s) \\biggr\\}  = 0.\n",
    "\\end{equation}\n",
    "\n",
    "$$ V(T,s) = \\Phi(s) $$\n",
    "\n",
    "The numerical algorithm for solving this PDE is almost identical to the algorithm proposed in the notebook **2.1** for the Black-Scholes PDE. \n",
    "\n",
    "Let us have a look at the differences between the implementation of the European and the American algorithm, in the class `BS_pricer`:\n",
    "\n",
    "```python\n",
    "if self.exercise==\"European\":        \n",
    "    for i in range(Ntime-2,-1,-1):\n",
    "        offset[0] = a * V[0,i]  \n",
    "        offset[-1] = c * V[-1,i]\n",
    "        V[1:-1,i] = spsolve( D, (V[1:-1,i+1] - offset) )\n",
    "elif self.exercise==\"American\":\n",
    "    for i in range(Ntime-2,-1,-1):\n",
    "        offset[0] = a * V[0,i]\n",
    "        offset[-1] = c * V[-1,i]\n",
    "        V[1:-1,i] = np.maximum( spsolve( D, (V[1:-1,i+1] - offset) ), Payoff[1:-1])\n",
    "```\n",
    "\n",
    "The European and the American algorithms just differ in one line!!    \n",
    "\n",
    "I don't consider dividends in the notebook. In absence of dividends the American call has the same value of the European call. For this reason, I'm going to consider only the pricing problem of a put option\n",
    "\n",
    "$$ \\Phi(S_{\\tau}) = ( K - S_{\\tau} )^+ $$\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "from FMNM.BS_pricer import BS_pricer\n",
    "from FMNM.Parameters import Option_param\n",
    "from FMNM.Processes import Diffusion_process\n",
    "from FMNM.cython.solvers import PSOR\n",
    "\n",
    "import numpy as np\n",
    "import scipy.stats as ss\n",
    "\n",
    "import matplotlib.pyplot as plt\n",
    "\n",
    "%matplotlib inline\n",
    "from IPython.display import display\n",
    "import sympy\n",
    "\n",
    "sympy.init_printing()\n",
    "\n",
    "\n",
    "def display_matrix(m):\n",
    "    display(sympy.Matrix(m))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Creates the object with the parameters of the option\n",
    "opt_param = Option_param(S0=100, K=100, T=1, exercise=\"American\", payoff=\"put\")\n",
    "# Creates the object with the parameters of the process\n",
    "diff_param = Diffusion_process(r=0.1, sig=0.2)\n",
    "# Creates the pricer object\n",
    "BS = BS_pricer(opt_param, diff_param)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The class `BS_pricer` implements two algorithms to price American options:\n",
    "- the Longstaff-Schwartz Method (see section [below](#sec2))\n",
    "- the PDE method"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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",
      "text/latex": [
       "$\\displaystyle 4.8181988706184$"
      ],
      "text/plain": [
       "4.818198870618403"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "BS.LSM(N=10000, paths=10000, order=2)  # Longstaff Schwartz Method"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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",
      "text/latex": [
       "$\\displaystyle 4.81563894941683$"
      ],
      "text/plain": [
       "4.815638949416825"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "N_space = 8000\n",
    "N_time = 5000\n",
    "BS.PDE_price((N_space, N_time))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Plot American put curve"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "<Figure size 900x500 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "fig = plt.figure(figsize=(9, 5))\n",
    "BS.plot([70, 140, 0, 30])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<a id='sec1'></a>\n",
    "## Optimal early exercise boundary\n",
    "\n",
    "Let us define a free boundary by the value $0<S_f<K$ such that if the current stock price $S_t$ is smaller than $S_f$, then it is optimal to exercise the option, otherwise it is optimal to wait.\n",
    "\n",
    "- $S_t < S_f \\quad $ called  **stopping region**\n",
    "- $S_t > S_f \\quad $ called  **continuation region**\n",
    "\n",
    "In the stopping region, the value of the option corresponds to its [intrinsic value](https://en.wikipedia.org/wiki/Intrinsic_value_(finance)) i.e. $V(t,S_t) = K-S_t$.     \n",
    "\n",
    "In order to find $S_f$ we have to find the maximum value $s$ such that $V(t,s) - (K-s) = 0$:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [],
   "source": [
    "payoff = BS.payoff_f(BS.S_vec).reshape(len(BS.S_vec), 1)  # Transform the payoff in a column vector\n",
    "mesh = (BS.mesh - payoff)[1:-1, :-1]  # I remove the boundary terms\n",
    "optimal_indeces = np.argmax(np.abs(mesh) > 1e-10, axis=0)  # I introduce the error 1e-10\n",
    "T_vec = np.linspace(0, BS.T, N_time)  # Time vector"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "<Figure size 900x500 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "fig = plt.figure(figsize=(9, 5))\n",
    "plt.plot(T_vec[:-1], BS.S_vec[optimal_indeces])\n",
    "plt.text(0.2, 94, \"Continuation region\", fontsize=15)\n",
    "plt.text(0.6, 87, \"Stopping region\", fontsize=16)\n",
    "plt.xlabel(\"Time\")\n",
    "plt.ylabel(\"$S_f$\")\n",
    "plt.title(\"Early exercise curve\")\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<a id='sec2'></a>\n",
    "## Binomial tree\n",
    "\n",
    "One of the most used methods for pricing American options is the Binomial tree. \n",
    "\n",
    "We have already encountered the binomial tree in the notebook **1.1**. The following algorithm is almost a copy/paste from that notebook.     \n",
    "There are just two additional lines:\n",
    "- `S_T = S_T * u`. This an efficient method to retrive the price vector at each time steps.  \n",
    "- `V = np.maximum( V, K-S_T )`. This line computes the maximum between the conditional expectation V and the intrisic value. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [],
   "source": [
    "S0 = 100.0  # spot stock price\n",
    "K = 100.0  # strike\n",
    "T = 1.0  # maturity\n",
    "r = 0.1  # risk free rate\n",
    "sig = 0.2  # diffusion coefficient or volatility"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "American BS Tree Price:  4.81624866310944\n"
     ]
    }
   ],
   "source": [
    "N = 25000  # number of periods or number of time steps\n",
    "payoff = \"put\"  # payoff\n",
    "\n",
    "dT = float(T) / N  # Delta t\n",
    "u = np.exp(sig * np.sqrt(dT))  # up factor\n",
    "d = 1.0 / u  # down factor\n",
    "\n",
    "V = np.zeros(N + 1)  # initialize the price vector\n",
    "S_T = np.array([(S0 * u**j * d ** (N - j)) for j in range(N + 1)])  # price S_T at time T\n",
    "\n",
    "a = np.exp(r * dT)  # risk free compound return\n",
    "p = (a - d) / (u - d)  # risk neutral up probability\n",
    "q = 1.0 - p  # risk neutral down probability\n",
    "\n",
    "if payoff == \"call\":\n",
    "    V[:] = np.maximum(S_T - K, 0.0)\n",
    "elif payoff == \"put\":\n",
    "    V[:] = np.maximum(K - S_T, 0.0)\n",
    "\n",
    "for i in range(N - 1, -1, -1):\n",
    "    V[:-1] = np.exp(-r * dT) * (p * V[1:] + q * V[:-1])  # the price vector is overwritten at each step\n",
    "    S_T = S_T * u  # it is a tricky way to obtain the price at the previous time step\n",
    "    if payoff == \"call\":\n",
    "        V = np.maximum(V, S_T - K)\n",
    "    elif payoff == \"put\":\n",
    "        V = np.maximum(V, K - S_T)\n",
    "\n",
    "print(\"American BS Tree Price: \", V[0])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<a id='sec3'></a>\n",
    "## Longstaff - Schwartz Method\n",
    "\n",
    "This is a Monte Carlo algorithm proposed by Longstaff and Schwartz in the paper [1]:\n",
    "[LS Method](https://people.math.ethz.ch/~hjfurrer/teaching/LongstaffSchwartzAmericanOptionsLeastSquareMonteCarlo.pdf)\n",
    "\n",
    "The algorithm is not difficult to implement, but it can be difficult to understand.  I think this is the reason the authors started the paper with an example.    \n",
    "Well. I think they had a good idea, and for this reason I want to reproduce their example here. \n",
    "\n",
    "The same code is copied in the class `BS_pricer` where the function `LSM` is implemented.\n",
    "\n",
    "**If in the following code you feel that something is unclear, I suggest you to follow the steps (not in python) proposed in the original paper**."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [],
   "source": [
    "N = 4  # number of time steps\n",
    "r = 0.06  # interest rate\n",
    "K = 1.1  # strike\n",
    "T = 3  # Maturity\n",
    "\n",
    "dt = T / (N - 1)  # time interval\n",
    "df = np.exp(-r * dt)  # discount factor per time interval"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [],
   "source": [
    "S = np.array(\n",
    "    [\n",
    "        [1.00, 1.09, 1.08, 1.34],\n",
    "        [1.00, 1.16, 1.26, 1.54],\n",
    "        [1.00, 1.22, 1.07, 1.03],\n",
    "        [1.00, 0.93, 0.97, 0.92],\n",
    "        [1.00, 1.11, 1.56, 1.52],\n",
    "        [1.00, 0.76, 0.77, 0.90],\n",
    "        [1.00, 0.92, 0.84, 1.01],\n",
    "        [1.00, 0.88, 1.22, 1.34],\n",
    "    ]\n",
    ")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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",
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}1.0 & 1.09 & 1.08 & 1.34\\\\1.0 & 1.16 & 1.26 & 1.54\\\\1.0 & 1.22 & 1.07 & 1.03\\\\1.0 & 0.93 & 0.97 & 0.92\\\\1.0 & 1.11 & 1.56 & 1.52\\\\1.0 & 0.76 & 0.77 & 0.9\\\\1.0 & 0.92 & 0.84 & 1.01\\\\1.0 & 0.88 & 1.22 & 1.34\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "⎡1.0  1.09  1.08  1.34⎤\n",
       "⎢                     ⎥\n",
       "⎢1.0  1.16  1.26  1.54⎥\n",
       "⎢                     ⎥\n",
       "⎢1.0  1.22  1.07  1.03⎥\n",
       "⎢                     ⎥\n",
       "⎢1.0  0.93  0.97  0.92⎥\n",
       "⎢                     ⎥\n",
       "⎢1.0  1.11  1.56  1.52⎥\n",
       "⎢                     ⎥\n",
       "⎢1.0  0.76  0.77  0.9 ⎥\n",
       "⎢                     ⎥\n",
       "⎢1.0  0.92  0.84  1.01⎥\n",
       "⎢                     ⎥\n",
       "⎣1.0  0.88  1.22  1.34⎦"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "display_matrix(S)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "In the previous cell we defined the stock matrix S.   \n",
    "It has 8 rows that correspond to the number of paths.  \n",
    "The 4 columns correspond to the 4 time steps, i.e. each row is a path with 4 time steps."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Example price=  0.11443433004505696\n"
     ]
    }
   ],
   "source": [
    "H = np.maximum(K - S, 0)  # intrinsic values for put option\n",
    "V = np.zeros_like(H)  # value matrix\n",
    "V[:, -1] = H[:, -1]\n",
    "\n",
    "# Valuation by LS Method\n",
    "for t in range(N - 2, 0, -1):\n",
    "    good_paths = H[:, t] > 0  # paths where the intrinsic value is positive\n",
    "    # the regression is performed only on these paths\n",
    "\n",
    "    rg = np.polyfit(S[good_paths, t], V[good_paths, t + 1] * df, 2)  # polynomial regression\n",
    "    C = np.polyval(rg, S[good_paths, t])  # evaluation of regression\n",
    "\n",
    "    exercise = np.zeros(len(good_paths), dtype=bool)  # initialize\n",
    "    exercise[good_paths] = H[good_paths, t] > C  # paths where it is optimal to exercise\n",
    "\n",
    "    V[exercise, t] = H[exercise, t]  # set V equal to H where it is optimal to exercise\n",
    "    V[exercise, t + 1 :] = 0  # set future cash flows, for that path, equal to zero\n",
    "    discount_path = V[:, t] == 0  # paths where we didn't exercise\n",
    "    V[discount_path, t] = V[discount_path, t + 1] * df  # set V[t] in continuation region\n",
    "\n",
    "V0 = np.mean(V[:, 1]) * df  # discounted expectation of V[t=1]\n",
    "print(\"Example price= \", V0)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The matrix `H = np.maximum(K - S, 0)`, is the matrix of intrinsic values:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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",
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}0.1 & 0.01 & 0.02 & 0\\\\0.1 & 0 & 0 & 0\\\\0.1 & 0 & 0.03 & 0.07\\\\0.1 & 0.17 & 0.13 & 0.18\\\\0.1 & 0 & 0 & 0\\\\0.1 & 0.34 & 0.33 & 0.2\\\\0.1 & 0.18 & 0.26 & 0.09\\\\0.1 & 0.22 & 0 & 0\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "⎡0.1  0.01  0.02   0  ⎤\n",
       "⎢                     ⎥\n",
       "⎢0.1   0     0     0  ⎥\n",
       "⎢                     ⎥\n",
       "⎢0.1   0    0.03  0.07⎥\n",
       "⎢                     ⎥\n",
       "⎢0.1  0.17  0.13  0.18⎥\n",
       "⎢                     ⎥\n",
       "⎢0.1   0     0     0  ⎥\n",
       "⎢                     ⎥\n",
       "⎢0.1  0.34  0.33  0.2 ⎥\n",
       "⎢                     ⎥\n",
       "⎢0.1  0.18  0.26  0.09⎥\n",
       "⎢                     ⎥\n",
       "⎣0.1  0.22   0     0  ⎦"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "display_matrix(H.round(2))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The matrix V contains the cash flows.\n",
    "\n",
    "**Important**    \n",
    "To simplify the computations, the discounted cashflows are reported at every time steps. \n",
    "\n",
    "For instance:    \n",
    "In the third row the final cashflow (0.07) is discounted at every time step, till t=1.    \n",
    "In the paper, the authors just consider the cashflow (0.07) at time t=3 and the discount is performed at the end of the algorithm."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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",
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0\\\\0 & 0.0621 & 0.0659 & 0.07\\\\0 & 0.17 & 0 & 0\\\\0 & 0 & 0 & 0\\\\0 & 0.34 & 0 & 0\\\\0 & 0.18 & 0 & 0\\\\0 & 0.22 & 0 & 0\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "⎡0    0       0      0  ⎤\n",
       "⎢                       ⎥\n",
       "⎢0    0       0      0  ⎥\n",
       "⎢                       ⎥\n",
       "⎢0  0.0621  0.0659  0.07⎥\n",
       "⎢                       ⎥\n",
       "⎢0   0.17     0      0  ⎥\n",
       "⎢                       ⎥\n",
       "⎢0    0       0      0  ⎥\n",
       "⎢                       ⎥\n",
       "⎢0   0.34     0      0  ⎥\n",
       "⎢                       ⎥\n",
       "⎢0   0.18     0      0  ⎥\n",
       "⎢                       ⎥\n",
       "⎣0   0.22     0      0  ⎦"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "display_matrix(V.round(4))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<a id='sec4'></a>\n",
    "## Perpetual put\n",
    "\n",
    "\n",
    "An American option is called *perpetual* if it has no expiration date i.e. $T=\\infty$:\n",
    "\n",
    "$$ V(t,s) =  \\sup_{\\tau \\in [t,\\infty]} \\mathbb{E}^{\\mathbb{Q}}\\biggl[ e^{-r(\\tau-t)}\\, ( K - S_{\\tau} )^+ \\, \\bigg|\\, S_t=s \\biggr]. $$\n",
    "\n",
    "By the time homogeneity of the problem we can set $t=0$ and obtain:\n",
    "\n",
    "$$ V(s) =  \\sup_{\\tau \\in [0,\\infty]} \\mathbb{E}^{\\mathbb{Q}}\\biggl[ e^{-r \\tau}\\, ( K - S_{\\tau} )^+ \\,\\bigg|\\, S_0=s \\biggr]. $$\n",
    "\n",
    "The PDE for the perpetual put option is:\n",
    "\n",
    "$$\n",
    "\\max \\biggl\\{   \n",
    "          r\\,s \\frac{ d V(s)}{d s}\n",
    "          + \\frac{1}{2} \\sigma^2 s^2 \\frac{d^2  V(s)}{d s^2} - r  V(s), \\;  \n",
    "          ( K - s )^+ - V(s) \\biggr\\}  = 0.\n",
    "$$\n",
    "\n",
    "In order to solve this obstacle problem we need to find a value $s_0 \\in (0,K)$ such that:\n",
    "\n",
    "$$ V(s) = ( K - s ) \\quad \\text{for} \\quad s \\in [0,s_0] \\quad \\text{ (stopping region)}$$\n",
    "\n",
    "and \n",
    "\n",
    "$$ r\\,s \\frac{ d V(s)}{d s}\n",
    "          + \\frac{1}{2} \\sigma^2 s^2 \\frac{d^2  V(s)}{d s^2} - r  V(s) = 0 \\quad \\text{for} \\quad s \\in [s_0, \\infty) \\quad \\text{ (continuation region)}$$\n",
    "          \n",
    "We can search for the solution of the linear second order ODE in the continuation region, by guessing a solution in the form $V(s) = s^n$.     \n",
    "We then obtain a quadratic equation in $n$ that has two solutions $n_1=1$ and $n_2 = -\\frac{2r}{\\sigma^2}$.     \n",
    "We can express \n",
    "\n",
    "$$V(s) = A s + B s^{-\\frac{2r}{\\sigma^2}} \\quad \\text{for} \\quad s \\in [s_0, \\infty).$$  \n",
    "\n",
    "Since $V(s) \\to 0$ as $s\\to \\infty$, we must have $A=0$.    \n",
    "We then require that *V(s)* is **continuous** and **differentiable** at $s_0$:\n",
    "\n",
    "$$ B s_0^{-\\frac{2r}{\\sigma^2}} = K -s_0 \\quad \\text{and} \\quad -\\frac{2r}{\\sigma^2} B s_0^{-\\frac{2r}{\\sigma^2}-1}  = -1$$\n",
    "\n",
    "These equations are satisfied for:\n",
    "\n",
    "$$ B = \\frac{\\sigma^2}{2r}\\, s_0^{\\frac{2r}{\\sigma^2}+1} \\quad \\text{and} \\quad s_0 = \\frac{K}{\\frac{\\sigma^2}{2r}+1}. $$\n",
    "\n",
    "It is also interesting to have a look at [2], where the author derives the formulas above using a probabilistic approach."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [],
   "source": [
    "def perpetual_put(S, K, sigma, r):\n",
    "    \"\"\"analytical formula for perpetual put option\"\"\"\n",
    "    s0 = K / (sigma**2 / (2 * r) + 1)\n",
    "    B = sigma**2 / (2 * r) * s0 ** ((2 * r) / sigma**2 + 1)\n",
    "    return np.where(S < s0, K - S, B * S ** (-2 * r / sigma**2))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [],
   "source": [
    "S = np.linspace(60, 160, 1000)  # stock price\n",
    "K = 100.0  # strike\n",
    "r = 0.1  # risk free rate\n",
    "sig = 0.2  # diffusion coefficient or volatility\n",
    "s0 = K / (sig**2 / (2 * r) + 1)  # optimal excercise value"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "<Figure size 1200x500 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "fig = plt.figure(figsize=(12, 5))\n",
    "plt.plot(S, perpetual_put(S, K, sig, r), label=\"perpetual option price\")\n",
    "plt.plot(S, np.maximum(K - S, 0), label=\"option intrinsic value\")\n",
    "plt.axvline(x=s0, color=\"k\", linestyle=\"--\", label=\"optimal excercise value\")\n",
    "plt.xlabel(\"s\")\n",
    "plt.ylabel(\"V(s)\")\n",
    "plt.title(\"Perpetual put option\")\n",
    "plt.legend()\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Numerical computation of perpetual put"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can pass to the log-variable $x = \\log s$ and obtain the simpler equation with constant coefficients:\n",
    "\n",
    "$$ \\max \\biggl\\{   \n",
    "          (r - \\frac{1}{2} \\sigma^2) \\, \\frac{ d V(x)}{d x}\n",
    "          + \\frac{1}{2} \\sigma^2 \\frac{d^2  V(x)}{d x^2} - r  V(x), \\;  \n",
    "          ( K - e^x )^+ - V(x) \\biggr\\}  = 0. $$\n",
    "\n",
    "equivalent to:\n",
    "\n",
    "$$ \\min \\biggl\\{   \n",
    "          -(r - \\frac{1}{2} \\sigma^2) \\, \\frac{ d V(x)}{d x}\n",
    "          - \\frac{1}{2} \\sigma^2 \\frac{d^2  V(x)}{d x^2} + r  V(x), \\;  \n",
    "          V(x) - ( K - e^x )^+ \\biggr\\}  = 0. $$\n",
    "\n",
    "This **variational inequality** can be reformulated as a **linear complementarity problem**:\n",
    "\n",
    "$$ \\biggl(   -(r - \\frac{1}{2} \\sigma^2) \\, \\frac{ d V(x)}{d x}\n",
    "          - \\frac{1}{2} \\sigma^2 \\frac{d^2  V(x)}{d x^2} + r  V(x),  \n",
    "          \\biggl)\\, \\cdot \\biggl( V(x) - ( K - e^x )^+ \\biggr)  = 0, $$\n",
    "$$ -(r - \\frac{1}{2} \\sigma^2) \\, \\frac{ d V(x)}{d x}\n",
    "          - \\frac{1}{2} \\sigma^2 \\frac{d^2  V(x)}{d x^2} + r  V(x) \\geq 0, \n",
    "          \\quad \\quad  V(x) - ( K - e^x )^+ \\geq 0.\n",
    "          $$          \n",
    "          \n",
    "At this poins, we can discretize the equations by finite difference using a central approximation for the first order derivative;\n",
    "\n",
    "$$  -(r - \\frac{1}{2} \\sigma^2)  \\frac{V_{i+1} -V_{i-1}}{ 2 \\Delta x} \n",
    "    - \\frac{1}{2} \\sigma^2  \\frac{V_{i+1} + V_{i-1} - 2 V_{i}}{\\Delta x^2} + r  V_i  \\geq 0 \n",
    " \\quad \\quad  V_i - ( K - e^{x_i} )^+ \\geq 0. $$\n",
    "\n",
    "That in matrix form becomes:\n",
    "\n",
    "$$  A \\, V - B \\geq 0 \\quad \\quad V - C \\geq 0 \\quad \\quad (A \\, V - B)\\cdot(V-C) = 0. $$\n",
    "\n",
    "where as usual, $A$ is a tridiagonal matrix and $B$ is the vector containing the boundary values. $C$ is the payoff.\n",
    "\n",
    "A description of numerical methods to solve the linear complementarity problem can be found in [3] or [4].     \n",
    "Here it is important to choose a big value of `S_max` because the perpetual put price goes to zero asimptotically.     \n",
    "The PSOR method is a slight modification of the SOR method introduced in the notebooks **A1** and **A2**. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [],
   "source": [
    "Nspace = 10000  # space steps\n",
    "S_max = 10 * K  # very large in order to minimize the error due to asymptotic boundary condition\n",
    "S_min = K / 3\n",
    "x_max = np.log(S_max)\n",
    "x_min = np.log(S_min)\n",
    "x, dx = np.linspace(x_min, x_max, Nspace, retstep=True)  # space discretization"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [],
   "source": [
    "sig2 = sig * sig\n",
    "dxx = dx * dx\n",
    "\n",
    "a = (r - 0.5 * sig2) / (2 * dx) - sig2 / (2 * dxx)  # -1 diagonal\n",
    "b = sig2 / dxx + r  # main diagonal\n",
    "c = -(r - 0.5 * sig2) / (2 * dx) - sig2 / (2 * dxx)  # +1 diagonal"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {},
   "outputs": [],
   "source": [
    "C = np.maximum(K - np.exp(x), 0)  # payoff\n",
    "\n",
    "B = np.zeros(Nspace)  # vector to be used for the boundary terms\n",
    "B[0] = -a * (K - S_min)  # boundary condition\n",
    "B[-1] = 0  # boundary condition"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Convergence after 30017 iterations\n"
     ]
    }
   ],
   "source": [
    "V = np.asarray(PSOR(a, b, c, B, C, w=1.999, eps=1e-12, N_max=1000000))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Plot"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "<Figure size 1600x500 with 2 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "fig = plt.figure(figsize=(16, 5))\n",
    "ax1 = fig.add_subplot(121)\n",
    "ax2 = fig.add_subplot(122)\n",
    "ax1.plot(S[::40], perpetual_put(S[::40], K, sig, r), \"*\", label=\"Theoretical price\")\n",
    "ax1.plot(np.exp(x), V, label=\"Numerical value\")\n",
    "ax1.plot(np.exp(x), C, label=\"Payoff\")\n",
    "ax1.axvline(x=s0, color=\"k\", linestyle=\"--\", label=\"optimal excercise value\")\n",
    "ax1.axis([30, 200, -1, 70])\n",
    "ax1.set_title(\"Numerical computation - Perpetual put option\")\n",
    "ax1.set_xlabel(\"s\")\n",
    "ax1.set_ylabel(\"V(s)\")\n",
    "ax1.legend()\n",
    "ax2.plot(np.exp(x), perpetual_put(np.exp(x), K, sig, r) - V, label=\"error\")\n",
    "ax2.set_title(\"Difference: theoretical - numerical\")\n",
    "ax2.set_xlabel(\"s\")\n",
    "ax1.set_ylabel(\"Error\")\n",
    "ax2.legend()\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## References\n",
    "\n",
    "[1] F. Longstaff, E. Schwartz (2001) *Valuing American Options by Simulation: A Simple Least-Squares Approach*, The Review of Financial Studies, vol 14-1, pag 113-147.   \n",
    "\n",
    "[2] Steven Shreve, *Stochastic calculus for finance, volume 2*, (2004).\n",
    "\n",
    "[3]  Daniel Sevcovic, Beata Stehlikova, Karol Mikula (2011). \"Analytical and numerical methods for pricing financial derivatives\". Nova Science Pub Inc; UK. \n",
    "\n",
    "[4] Wilmott Paul (1994). \"Option pricing - Mathematical models and computation\". Oxford Financial Press."
   ]
  }
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